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IV. On the Permanent State of Heat in a Thin Uniform Wire 

 of any form, acted on by two sources of heat of equal inten- 

 sity at its extremities. By G. W. Hearn, B.A. of Cam- 

 bridge, and Professor of Mathematics, Royal Military Col- 

 lege, Sandhurst*. 



WE suppose the wire homogeneous and of the same 

 thickness throughout, s 2 the area of a transverse sec- 

 tion, K the interior, and h the exterior conductivity ; v the 

 temperature at a point distant s from one extremity measured 

 along the wire. 



The excess of the flux of heat between s and s + 8 s, is 



ds* 

 and the quantity emitted by this portion is 



h c v 8 s, 

 c being the circumference of the transverse section, 



Ks z -j-^—hcv, 

 or if g^ = h\ 



ke^Js, 



H = ^ 



ds> 



.•. v = A e~ ks + B e ks . 

 Let the equal temperatures of the extremities be repre- 

 sented by 1, and let a be the length of the wire; 

 .-. A+ B= 1, 

 A e~ ka + B e ka = 1, 



. e ka — 1 _, 1— e~ ka 

 or A = -r r , B = —, r » 



fika o—ku? pica a —ka' 



v= A{e- ks + e- k ^ a - s ^} (1.) 



Now suppose the temperature at the middle point to have 

 been determined by observation = t r 



r ka ka~\ ka 



Then t% = A \e~* + e~~*~ J* =2A e", 



oka i ka Q 



or L = 2-7- =- . e~~5~= 



,ka__g~ka' " ka ka* 



e~2 + e~~z~ 

 from which 



*5 1+ Vl— jf 2 2 

 ea = — • 



* Communicated by the Author. 



